Angle-bisector Proof

Date: 10/16/97 at 14:46:12
From: Tim
Subject: Geometry
Dear Dr. Math,
Prove that in a triangle ABC, a pair of angle-bisectors cannot be
perpendicular.
I have absolutely no idea what this is even about. Haven't done
geometry in 15 years!
Yours truly,
Tim in Montreal

Date: 10/16/97 at 15:47:35
From: Doctor Wilkinson
Subject: Re: Geometry
Draw a picture!
Suppose the bisector of the angle at A and the bisector of the angle
at B are perpendicular. Suppose they intersect at D. This gives you a
smaller triangle ABD, where the angle at D is a right angle. That
tells you that the angle DAB and the angle DBA add up to 90 degrees,
because the angles of a triangle always add up to 180 degrees. That
means that the angle CAB, which is twice DAB (because AD is the angle
bisector; and the angle CBA, which is twice DBX (Because BD is the
angle bisector), add up to twice 90 degrees, or 180 degrees. But
that's impossible, because all three angles of the triangle have to
add up to 180 degrees, and there's nothing left.
-Doctor Wilkinson, The Math Forum
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